Algebra - Washington State University
Algebra
1. (a + b)2 = a2 + 2ab + b2 2. (a - b)2 = a2 - 2ab + b2 3. a2 - b2 = (a - b)(a + b)
4. ax2 + bx + c =
-b + b2 - 4ac a x-
2a
if b2 - 4ac 0
-b - b2 - 4ac
x-
,
2a
Trigonometry
Definitions
sin x 1. tan x =
cos x cos x 2. cot x = sin x
Identities
1. cos2 x + sin2 x = 1
2. sin 2x = 2 sin x cos x
3. cos 2x = = cos2 x - sin2 x = 2 cos2 x - 1 = 1 - 2 sin2 x
1 3. sec x =
cos x 1
4. csc x = sin x
4. tan2 x + 1 = sec2 x 5. cot2 x + 1 = csc2 x
2 tan x 6. tan 2x = 1 - tan2 x
cot2 x - 1 7. cot 2x =
2 cot x
Hyperbolic functions
Definitions
ex - e-x
1. sinh x = 2
1 3. csch x =
sinh x sinh x 5. tanh x = cosh x
ex + e-x
2. cosh x = 2
1 4. sech x =
cosh x cosh x 6. coth x = sinh x
Identities 1. cosh2 x - sinh2 x = 1
4. sinh 2x = 2 sinh x cosh x
2. 1 - tanh2 x = sech2 x 3. coth2 x - 1 = csch2 x
5. cosh 2x = = cosh2 x + sinh2 x = 2 cosh2 x - 1 = 2 sinh2 x + 1
Derivatives
Derivatives
1. (sin x) = cos x 2. (cos x) = - sin x 3. (tan x) = sec2 x 4. (cot x) = - csc2 x
1 7. (arcsin x) =
1 - x2 -1 8. (arccos x) = 1 - x2
5. (sec x) = tan x sec x 1
6. (csc x) = - cot x csc x 9. (arctan x) = 1 + x2
Integrals
1. (sinh x) = cosh x
2. (cosh x) = sinh x 3. (tanh x) = sech2 x 4. (coth x) = - csch2 x
5. (sech x) = = - tanh x sech x
6. (csch x) = = - coth x csch x
7. (arcsinh x) = 1
= x2 + 1
8. (arccosh x) = 1
= x2 - 1
9. (arctanh x) = 1
= x2 - 1
1. cos x dx = sin x
6. tan x dx =
2. sin x dx = - cos x
= - ln | cos x|
3. sec2 x dx = tan x
7. sec x dx = = ln | tan x + sec x|
4. csc2 x dx = - cot x 8. csc x dx =
5. cot x dx = ln | sin x|
= - ln | cot x + csc x|
Symmetry & Periodicity 1. sin(-x) = - sin(x) 2. sin(x + /2) = cos(x) 3. sin(x - /2) = - cos(x) 4. sin(x ? ) = - sin(x)
5. cos(-x) = cos(x) 6. cos(x + /2) = - sin(x) 7. cos(x - /2) = sin(x) 8. cos(x ? ) = - cos(x)
9. tan(-x) = - tan(x) 10. tan(x + /2) = - cot(x) 11. tan(x - /2) = - cot(x) 12. tan(x ? ) = tan(x)
x
- -/2 0 /6 /4 /3 /2 2/3 5/6 3/2 2
sin(x) 0
-1 0
1 2
1 2
3 2
1
3 2
1 2
0 -1 0
cos(x) -1
0
1
3 2
1 2
1 2
0
-
1 2
-
3 2
-1
0
1
tan(x) 0
?
0 1
1
3
?
-3
- 1
0
?
?
3
3
cot(x) ?
0
?3
1
1
1
- 1
-3
?
0
1
3
3
Powers, Exponents & Logarithms
1. xa+b = xa ? xb
2.
xa-b =
xa xb
3.
x-a =
1 xa
4. xa?b = (xa)b
5. xa/b = b xa
6. ex : (-, ) (0, ) 7. eln(x) = x 8. ln(ex) = x
Derivatives & Integrals 1. (xn) = nxn-1
3. (eax) = aeax
2.
xn
dx =
xn+1 ,
n = -1
n+1
4. eax dx = eax
a
9. ln(x) : (0, ) (-, )
10. ln(a ? b) = ln(a) + ln(b) a
11. ln = ln(a) - ln(b) b
12. ln(ab) = b ln(a)
1 5. (ln(x)) =
x
dx
6.
= ln(x)
x
7. ln(x) dx = x ln(x) - x
Other Integrals
1
1
x
1. x2 + a2 dx = a arctan |a|
2. 1 dx = ln |x + x2 ? a2| x2 ? a2
Inverse trigonometric functions
arcsin x = arccos 1 - x2
sin(y) cos(y) tan(y) cot(y)
y = arcsin(x) x
1 - x2 x
1 - x2
1 - x2 x
y = arccos(x)
1 - x2
x
1 - x2 x x
1 - x2
y = arctan(x) x
x2 - 1 1
1 + x2
x
1 x
y = arccot(x) 1
1 + x2 x
x2 - 1 1 x
x
Inverse hyperbolic functions
sinh(y) cosh(y) tanh(y) coth(y)
y = arcsinh(x) x
x2 + 1 x
x2 + 1
x2 + 1 x
y = arccosh(x) x2 - 1
x
x2 - 1 x x
x2 - 1
y = arctanh(x) x
1 - x2 1
1 - x2
x
1 x
y = arccoth(x) sign x
x2 - 1 |x|
x2 - 1 1 x
x
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